PARTIAL DISTRIBUTION FUNCTION AND RADIAL STATISTICAL COEFFICIENTS FOR Be-ATOM

Transcription

1 Journal of Krbala Unvrsty, Vol. 7 o.4 Scntfc. 009 PARTIAL DISTRIBUTIO FUCTIO AD RADIAL STATISTICAL COEFFICIETS FOR B-ATOM Mohammd Abdulhussan Al-Kaab Dpartmnt of Physcs, collg of Scnc, Krbala Unvrsty ABSTRACT Approxmaton calculatons usng Hartr-Fock wav functon ar carrd out on th S ground stat for th Bryllum atom n poston spac.for ach shll n ths atom th partal dstrbuton functon, radal statstcal coffcnts, dnsty dstrbutons and xpctaton valus ar studd.ths proprts ar achvd from th parttond form of th dnsty dstrbuton. Th ffcts of th lctron spn n both ntr-and ntra-shll on th abov proprts s takn nto account. 173 الخالصة اجزيث في هذا البحث حسابات ججزيبية اسحخدهث فيها الدالة الوىجيةة لهةارجز فةى لدراسةة الوسةحى ارريةي لةذر البزيليىم في فضاء الوىقع. ولكل قشز هي قشز هذا ال ظةام جةن اراسةة االةة جىسيةع الكاافةة الجشليةة والولةاهتت السةحاجيكية الشةلا ية واالة جىيع الكاافة والقين الوحىقلة. جن ا جاسالخصةال الودروسةة ا ةت باسةححدام ةيتة الحجشلةة لحىسيةع الكاافةة فوقةد جةن فةي هةذ الدراسة ارخذ بليي ار حبار جاثيز بزم ارلكحزوى لجويع القشز والقشز الداخلية هع هقار ة ال حالج هع ال حالج اللالوية الوحاحة. ITRODUCTIO On of th frst and most smpl approxmatons amd to solv th problm ars from rpulsv of lctrons n many lctron systm s du to Hartr n 197 (1).In ths approxmaton,th total wav functon s constrand to b a product of on-lctron orbtals whr th spn of lctrons and th ant- symmtry ar not tak nto account. Th most proprty whch s nglctd by Hartr t s th ndstngushablty of th lctrons, and ths notcd ndpndntly by Fock () and Slatr (3) n 1930, and t was corrctd agan to taks th form of a so-calld Slatr dtrmnant (s quaton 1). Th mathmatcal consqunc of th ndstngushablty among a st of -quantum objct of th sam typ (such lctrons, tc) s th rqurmnt that th total -partcl wav functon aftr xchang any par of a partcls coordnats must thr rman unchangd (symmtrc), n ths cas, th partcls ar bosons and must hav ntgr spn, or chang sgn (ant-symmtrc) and a partcls hr calld frmons and hav sm-ntgr spn. Snc th lctrons ar frmons, thrfor, th total wav functon must b ant-symmtrc undr th xchang of any par of lctrons (4).Thus, ths proprty cannot satsfy by th Hartr product, but t s satsfy by lnar combnaton form, whch proposd by Slatr dtrmnant. Many studs dal wth th dstrbuton functon g (r 1, r1), Banyard and Mobbs (5) usd th HF and CI systms n thr studyng for ths functon and dtrmnd th partal coulomb hol Δg (r 1, r1)for B atom, Banyard and Rd (6) studd ths dstrbuton functon by usng of (FC) and HF wav functons for HH +,Al-Bayat and Banyard (7) usd th CI and HF wav functons through thr studyng of L-lk ons n frst xctd stat. Statstcal corrlatons btwn lctrons can b studs n many ways, Coopr and Poundr (8), Sharma and Thakkar (9) studd ths corrlaton by analyzng lctron corrlaton hols, Sandrs and Banyard (10), Wang t al (11) and Koga (1) xamnng th par corrlaton dnsty tslf. In th prsnt papr w apply th g (r 1, r1) and th da of Kutzlngg t al (13) for statstcal corrlaton to th ground stat of B-atom by usng of HF wav functon. To study ths proprts n dpth w usd portonng tchnqu and rprsnt th dstrbuton functon as a surfac and contour dagrams.

2 Journal of Krbala Unvrsty, Vol. 7 o.4 Scntfc. 009 THEORY AD WAVEFUCTTIO In th Hartr-Fock systm, th total ant-symmtrc wav functons wrttn as Slatr dtrmnant (14) :- x1,..., x 1x1. x,..., x, (1) In ths quaton th Drac notatons rprsnt th Slatr dtrmnant and th normalzaton constant, th on-lctron orbtal, n addton to, normalzd proprty, t s also mutually orthogonal (15) j j, j 1,,..., () Also, w usd on-lctron wav functon dpndnt also on th spn,thus, our wav functon s spn-orbtals. A gnral spn-orbtal may b wrttn as (4) :- x r r (3) Whr th functons and corrspondng to th spn-up and spn-down. Evry on-lctron orbtal functon r conssts of normalzd Slatr typ orbtals (16) :- l m R ry r, (4) nml l n n l n1 0.5 Whr R r r xp r, n!, n, l and m ar quantum numbrs and s calld xponnt paramtr. In many lctron systms, th Hamltonan s gvn by (17) :- 1 1 Hˆ Tˆ Vˆ Vˆ 1 Z r 1 1 r or 1 1 H ˆ h 1 1 r (6) j Whr r n quaton (5) s th dstanc of on lctron to th nuclus. Thn: H ˆ h 1 1 r (7) j Ths lads to:- Hˆ hˆ 1 1 r j (8) ow, w wll usful from th partal dstrbuton functon j j j ĥ to calculat th xpctaton valu for r and thn calculats th g r 1,r 1 and radal statstcal coffcnts. To satsfy our goal w wll us th parttonng tchnqu to studng ach two-lctron n on shll, and us th wav functon n trms of th radal two-partcl dnsty dstrbuton jr, r 1 whch obtand from (5) :- j r, r 1 x x x x (9) 1 1 j j 1 (5) By substtutng th quatons (3 and 4) n quaton (9) and ntgratng ovr spn and angular functon w can gt:- For ntra-shll K α K β (K( 1 S)) 1, j :- r r R r R (10) 1 1, 1s 1 1s r 174

3 Journal of Krbala Unvrsty, Vol. 7 o.4 Scntfc. 009 For ntr-shll KL( 3 S) 1, j 3 : s 1 R s 1s 1 s 1 s 1s s 1 1s r r r R r R r R r R r R r R R (11) For ntr-shll KL( 1 S) 1, j 4: R1s r1 Rs r Rs r1 R1s r (1) For th ntra-shll L( 1 S) 3, j 4 :- r r R r R 34 1, s 1 s r (13) ow th partal dstrbuton functon s gvn by :- r1r1 gr1, r1 1 r1, r r dr j r1 (14) r1r1 Whr r 1 s th dstanc btwn th two lctrons. For our two-lctron shlls, th radal statstcal coffcnts can b ntroducd as (18) :-, and (15) Such that th on-lctron xpctaton valu n (a.u.) s gvn by (19) :- 0 r1 r1 1 r1 dr (16) r 1 s on-partcl radal dnsty and dfnd as (18) :- r1 r1, r dr (17) Fnally, our paramtrs c, n, ar takn from atomc data and nuclar data tabls for Rothaan- Hartr-Fock ground stat wav functon for Bryllum whch ctd by Rf.(0),and th Mathcad program usd hr to solv th ntgratons. RESULTS AD DISSCUTIO Frstly, w can dfnd th dstrbuton functon g(r 1,r 1 ) to rprsnt th probablty of fndng an ntr-lctronc sparaton r 1 whn a tst lctron r 1 s locatd at a dstanc r 1 from th nuclus, whr th nuclus locatd at th cntr of coordnats. Also, th ntgraton ovr g HF (r 1,r 1 ) from 0 gvs th radal dstrbuton functon D HF (r 1 ) whch s normalzd to unty. Th functon g HF (r 1,r 1 ) s mportant to studyng th partal coulomb hols (Δg(r 1,r 1 )= g xact. (r 1,r 1 )- g HF (r 1,r 1 )),ths hols ar nabl us to xamn th ffct of lctron corrlaton whn a tst lctron s locatd at a spcfc dstanc from th nuclus. Fgur (1-a,b) rprsnts th contour and surfac dagrams bhavor of g HF (r 1,r 1 ) aganst (r 1,r 1 ) for K α K β shll, w s th man charactrstc fatur locatd about th dagonal ln whr r 1 = r 1 and ths fatur also paralll to th r 1 axs for small r 1. For surfac dagram, w s that th maxmum for g HF (r 1,r 1 ) =3.784 occurs at r 1 = r k (K α K β )=0.7,whr th r 1 rprsnt th mod of D HF (r 1 ) for th K-shll, ths man that th hghr probablty of radal dstrbuton functon D HF (r 1 ) occurs at r k (s fgur 5-a).At ths maxmum valu of g HF (r 1,r 1 ) w not that r 1 > r 1,t s found In contour dagram fgur (1-a) w can s ths bhavor, but th maxmum valu not appars hr bcaus w slct som contours to gt a good pctur. It s worth to mnton hr that th valu appar n ach contours rprsnt th valu of g HF (r 1,r 1 ) for fxd r 1 and varous r 1. For ntr-shlls KL( 3 S) (K α L α = K β L β ) whr th two lctrons wth sam spn drcton, fgur () show us a dffrnt bhavors of g HF (r 1,r 1 ),whr two man faturs appar nstad of on as n K-shll. Ths two faturs ar charactrstc of an ntr-shll dstrbuton (1). It s clar from th 175

4 Journal of Krbala Unvrsty, Vol. 7 o.4 Scntfc. 009 contour and surfac dagrams (3-a, b) that on of ths faturs l paralll to r 1 axs startng from small r 1 wth valu at r 1 =.04, r 1 = r k = 0.6 and th othr along th dagonal axs (r 1 =r 1 ) wth valu of at r 1 =.05 and r 1 = r L =.04. Also, th r 1 valus whch locatd th two maxmum valus of dstrbuton functon ar th local K and L-shll mods of D HF (r 1 ) (s fgur 5-b ).Bcaus of th sam spn componnts of our two lctrons th Frm ffcts a rass and th consqunc no valu of g HF (r 1,r 1 ) at small r 1 for all rangs of r 1. Th partal dstrbuton functon g HF (r 1,r 1 ) for ntr-shlls KL( 1 S) (K α L β =K β L α ) rprsntd by fgur(3-a,b),ths fgur shows faturs smlar to thos appard n prvous ntr- shlls but wth nw fatur at a dstanc clos to nuclus. At th dagonal (r 1 = r 1 ) and paralll axs w s that th maxmum dstrbuton s smlar to th prvous ntr-shlls n maxmum and locatons.snc th two lctrons wth oppost spn drcton Frm ffct wll not prsnt, hnc, addtonal fatur "mn-shll" wll appars clos to th nuclus. Ths mn-k-shll wth small valu of g HF (r 1,r 1 ) and sam n shap and locaton th fatur of K-shll, t s ars from pntratd of L- shll lctron to K-shll thn ntractd wth ts lctron as a consqunc mn-shll sn at a small r 1 and r 1. In th ntra-shll (L-shll) th two lctrons wth th oppost spn drcton, thn w s that th maxmum valu of functon g HF (r 1,r 1 ) s lss than thos for prvous ntr-shll, t s at r 1 =.05 and r 1 =.11. Fgur (4-a, b) show anothr probablty n ths shll locatd at small valu of axs, r 1 =0.1 and r 1 =0.43, th small probablty of L-shll a rass from attracton ntracton of our lctrons bcaus thr spns. For any ntrshlls thr s spatal ntracton btwn th two lctrons, ths ntracton socalld lctron corrlaton, t s mans that th moton of th lctrons ar statstcally corrlatd (18) (ths ffcts has bn partally takn nto account by th Hartr-Fock modl). Accordng to standard concpt for mathmatcal statstcs and probablty thory Kutzlngg t al. (13) usd th statstcal coffcnts as a prob for th analyss of lctron corrlaton n th atomc systm, dpndng on som xpctaton valus,th xpctaton valu and for n =- ar lstd n tabls (1,). Th spatal corrlaton can sn n th ntrshlls that occupd by two lctrons wth sam spn componnts or wth oppost componnts. In ths cass, th lctrons sparatd by a dstanc btwn ach othrs, thrfor, spatal corrlaton (statstcal corrlaton) wll a ras. So, for shlls (K or L) th lctrons n th sam shll, thrfor, must b wth oppost spn drctons as a rsult, thy ar suffrd from rpulsv ntracton, thn no chanc for spatal ntracton. Th last lads to all th for th HF wav functon ar zro xcpt for th radal coffcnts for ntrshlls. Tabl (3) show us zro valu of (K or L) shlls and non zro valus of ntrshlls. REFERECES 1. D.R.Hartr, Proc. Camb. Phl.soc., 4,89,111(197).. V. Z. Fock, Phys., 61,16(1930). 3. J. C. Slatr, Phys. Rv.,35,10(1930) K.E. Banyard and R. J. Mobbs, J. Chm.Phys.,75, o.7,3433(1981). 176

5 Journal of Krbala Unvrsty, Vol. 7 o.4 Scntfc K.E. Banyard and C. E. Rd, J. Phys. B:At. Mol. Phys., 14,411(1981). 7. K. H. AL-Bayat and K. E. Banyard J. Phys. B,14,411 (1986). 8. I. L. Coopr and C.. M. Poundr, Int. J. Quant. Chm., 17,759(1980) 9. B. S. Sharma and A. J. Thakkar, J. Phys. B, 17, 3405(1984). 10. J. Sandr and K. E. Banyard,J. Chm. Phys.,96,4535(199). 11. J. Wang, A.. Trpath and V. H. Jr. Smth, J. Phys. B, 6,05(1993). 1. T. Koga, J. Chm. Phys.,115,6847(001). 13. W. Kutzlngg, R. G. Dl and G. Brthr, Phys. Rv. A,47, 17(1968) (000) A. W. Wss, J. Chm.Phys.,39, 16(1963). 17. F. W. Kng, J. Mol. Strac.,400, 7(1997). 18. J. Wang and V. H. Smth, Tho. Chm. Acta., 88, 35(1994). 19. F.W. Kng and P. R. Drssl, J. Chm. Phys.,90,o. 11,6449(1989). 0. C.F. Bung and J. A. Barrntos, At, Data ucl.53,o.1,113 (1993). 1. E. Banyard and K. H. AL-Bayat, J. Phys. B:At. Mol. Phys., 19,11(1986).. K.E. Banyard and M. M. Mashat, J. Chm.Phys.,67,o.4,1405(1977). Tabl (1): Som on-partcl xpctaton valus for ntra-and ntr shlls of B-atom. Shll a.u. n =- n =-1 n =1 n = K( 1 S) a a a a KL( 1 S)=KL( 3 S) a.10 a a a L( 1 S) a 0.55 a.6498 a a Total c b.10 b b b a from rfrnc(5). b calculatd for rf.(5). c th total lstd hr s gvn by 1/6(K( 1 S)+L( 1 S)+KL( 1 S)+3KL( 3 S)) Tabl (): Som two-partcl xpctaton valus for ntra-and ntr shlls of B-atom. Shll K( 1 S) a.u. n =- n =-1 n =1 n = d 0.17 d

6 Journal of Krbala Unvrsty, Vol. 7 o.4 Scntfc. 009 KL( 3 S) d d KL( 1 S) d d L( 1 S) d from rfrnc () d d Total c d d Tabl (3): Som radal statstcal coffcnts for ntra-and ntr shlls of B-atom. Shll K( 1 S) KL( 3 S) KL( 1 S) L( 1 S) Total from Rf(5) 178

7 Journal of Krbala Unvrsty, Vol. 7 o.4 Scntfc. 009 Fgur (1): dstrbuton functon g HF (r 1, r1) aganst (r 1, r1) for K-shll contour dagram surfac dagram (scals for fgurs 1 4 dvdd on 0.05). Fgur (): dstrbuton functon g HF (r 1, r1) aganst (r 1, r1) for KL ( 3 S)-shll contour dagram surfac dagram. 179

8 Journal of Krbala Unvrsty, Vol. 7 o.4 Scntfc. 009 Fgur (3): dstrbuton functon g HF (r 1, r1) aganst (r 1, r1) for KL ( 1 S)-shll contour dagram surfac dagram. Fgur (4): dstrbuton functon g HF (r 1, r1) aganst (r 1, r1) for L-shll contour dagram surfac dagram. 180

9 Journal of Krbala Unvrsty, Vol. 7 o.4 Scntfc. 009 (c) (d) Fgur (5): On partcl radal dnsty aganst dstanc for ground stat of B-atom. K- shll. Intrshlls ( 3 S) (c) ntrshlls ( 1 S) (d) L-shll. 181